ARMA(p,q) Model

ARMA(p,q) Model

The ARMA(p,q) model combines AR and MA components:

y[t] = φ₁·y[t-1] + ... + φp·y[t-p] + ε[t] + θ₁·ε[t-1] + ... + θq·ε[t-q]

It is one of the most widely used models in time series analysis. ARMA models require stationarity — for non-stationary series, we use ARIMA (which adds differencing).

Simulation

To simulate ARMA(p,q), initialize with xs_init values and pad past errors with zeros. At each step:

1. AR part: weighted sum of last p y-values
2. MA part: current error plus weighted sum of last q errors
3. New value = AR part + MA part

One-step Forecast

For forecasting, the future error is 0 (expectation):

ŷ[t+1] = φ₁·y[t] + ... + φp·y[t-p+1] + θ₁·ε[t] + ... + θq·ε[t-q+1]

Task

Implement:

• arma_simulate(xs_init, phi, theta, errors, n)
• arma_forecast(xs, errors, phi, theta)